AVO-preserving processing

Quantitative interpretation (QI) workflows (AVO analysis, inversion for elastic attributes, pre-stack simultaneous inversion) all depend on relative amplitudes being preserved through the processing chain. A CMP's reflection amplitude as a function of incidence angle encodes the reflector's intercept and gradient, which in turn encode fluid content, lithology and porosity. Any processing step that rescales amplitude differently at each offset moves those two numbers. This section is about which steps do that, and what an amplitude-preserving flow looks like.

1. The AVO model, Shuey 2-term

For small contrasts and incidence angles up to about 30^\\circ, Aki-Richards reduces to the 2-term Shuey approximation:

where $A = \\tfrac{1}{2}(\\Delta V\_{\\mathrm P}/V\_{\\mathrm P} + \\Delta\\rho/\\rho)$ is the intercept (normal-incidence reflectivity) and $B = \\tfrac{1}{2}\\Delta V\_{\\mathrm P}/V\_{\\mathrm P} - 2(V\_{\\mathrm S}/V\_{\\mathrm P})^2(2\\Delta V\_{\\mathrm S}/V\_{\\mathrm S} + \\Delta\\rho/\\rho)$ is the gradient. The term the approximation drops, $C(\\tan^2\\theta - \\sin^2\\theta)$ with $C = \\tfrac{1}{2}\\Delta V\_{\\mathrm P}/V\_{\\mathrm P}$, grows quickly past 30^\\circ. Extracting $(A, B)$ per CMP is the core of AVO analysis; their joint distribution classifies reservoirs (Classes I to III per Rutherford and Williams 1989, IIp per Ross and Kinman 1995, Class IV per Castagna and Swan 1997).

2. Process a sand yourself

Figure 7.1 opens on a Class III gas sand whose spherical spreading was never corrected. Restore the correction with the gain exponent $n$, then switch on AGC or mute the far angles, and watch where the fitted $(\\hat A, \\hat B)$ lands against the rock's own $(A, B)$ in plate (d).

With no spreading correction the gas sand's gradient reads $\\hat B = -0.073$ against the rock's $-0.151$: the far angles come out dimmed by about $\\cos\\theta$, and a sand that should brighten with angle barely does. Restore $n = 1$ and the fit returns $-0.152 \\pm 0.016$. Amplitude-preserving processing returns $\\hat A$ and $\\hat B$ equal to the rock's within noise; AGC and a missing spreading correction bias them, and an aggressive mute leaves them unbiased but much noisier.

3. The four processing modes

• Amplitude-preserving. No AGC, spherical spreading corrected, no aggressive mute. The recovered $(\\hat A, \\hat B)$ equal the true $(A, B)$ within noise, apart from the small bias of the two-term form itself. This is the target.
• AGC applied. Automatic Gain Control scales each trace by the inverse of its amplitude in a sliding time window. It is widely used for visual interpretation because it makes events visible at all depths, but it destroys angle-dependent amplitude variation. For an isolated reflector the recovered gradient $\\hat B$ goes to zero whatever the true $B$ was: a brightening Class III anomaly becomes a flat negative event with no AVO at all. In Figure 7.1 a 500 ms window leaves $\\hat B = -0.028$ against $-0.151$, while $\\hat A$ stays negative. With other reflections in the window the gain depends on them too, so the damage cannot be predicted. If AGC appears anywhere in a QI flow, the flow is broken.
• Spherical spreading uncorrected. Spherical divergence is the geometric amplitude decay of a wavefront: energy spreads as $1/r^2$, so amplitude drops as $1/r$. It is corrected by multiplying by $t\\,V\_{\\mathrm{rms}}^2(t)$ in production. If that correction is missed, far-offset amplitudes are suppressed by roughly $t\_0/t(\\theta) \\approx \\cos\\theta$, so to first order $\\hat B \\approx B - A/2$: every reflector dims with angle, positive-intercept events gain a false negative gradient, and negative-intercept (Class III) events lose their brightening.
• Aggressive stretch mute. NMO stretches far-offset samples, reducing their effective frequency, and production flows mute samples stretched past a threshold. For a flat reflector the stretch $1/\\cos\\theta - 1$ depends only on the angle, so a stretch limit is an angle limit. If the mute is too aggressive (keeping only angles below 25^\\circ), you lose the far-offset samples that carry the gradient. $\\hat B$ does not become biased; it becomes noisy. In Figure 7.1 a 15^\\circ mute makes its standard error 5.0 times that of the full 30^\\circ aperture, for the same noise.

4. What an amplitude-preserving flow looks like

1. Source signature deconvolution. Remove the source wavelet carefully using measured far-field signatures where possible. Surface-consistent deconvolution equalises shot-receiver pairs without introducing arbitrary amplitude bias.
2. Spherical divergence correction. Multiply by $t\\,V\_{\\mathrm{rms}}^2(t)$ to compensate the geometric wavefront decay.
3. Surface-consistent amplitude (SCA) balancing. Decompose amplitude variations into shot, receiver, offset, and CDP terms, remove the shot and receiver terms that reflect acquisition non-uniformities, keep the offset and CDP terms that carry geology.
4. Static corrections applied without amplitude scaling: statics is a time shift, not an amplitude operation.
5. Ghost deconvolution for marine data, designed with the angle-dependent ghost operator; a vertical-incidence deghost leaves an offset-dependent amplitude error.
6. Demultiple (SRME + Radon with a moderate moveout cutoff): parameterise it more conservatively than for imaging, because Radon can remove primary energy at near offsets where moveout discrimination is poor.
7. Mild NMO with a small-stretch mute (typically a stretch limit that keeps angles to about 35^\\circ, not 25^\\circ).
8. No AGC anywhere. Anywhere.
9. Amplitude-preserving pre-stack time or depth migration (Section 7.2) as the final step before AVO extraction.

5. How to spot AVO distortion on real data

• Every reflector has a flat gradient ($\\hat B \\approx 0$) whatever its lithology. Likely: AGC somewhere upstream.
• Every reflector dims with angle (a false negative gradient on positive-intercept events). Likely: spherical spreading correction missing; if every reflector brightens with angle instead, it is over-corrected.
• The gradient is noisy and the far angles are missing. Likely: aggressive stretch mute.
• A known wet-sand reflector leaves the background trend. On the intercept-gradient crossplot, brine sands and shales fall along a background (wet) trend: their gradients are generally not zero but roughly anti-correlated with their intercepts (Castagna, Swan and Foster 1998). If a known wet sand plots off that trend, something in the flow is distorting amplitudes.

6. What to do if the flow was not amplitude-preserving

Reprocess. The damage from AGC cannot be undone unless its gain function was saved: the information is gone. A missing spreading correction is deterministic and can be applied later, but it is safer to rebuild the flow. Occasionally a flow can be salvaged by stacking across a single reflector and back-inverting for a relative correction, but the cleaner and more defensible answer is to rebuild the pre-stack data from raw and apply an explicit amplitude-preserving flow from the start. Project managers resist this because reprocessing is expensive; the alternative is shipping a QI product that is inconsistent with geology, which is a worse outcome.

Where this goes next

Section 7.2 covers the migration step: Kirchhoff migration with proper amplitude weighting so that after migration, amplitudes still reflect the subsurface contrast rather than the operator's geometric aperture. This is "true-amplitude" migration, as opposed to the "structural" migration that is adequate for visual interpretation.

References

• Castagna, J. P., Backus, M. M. (1993). Offset-Dependent Reflectivity. SEG.
• Russell, B. H. (1988). Introduction to Seismic Inversion Methods. SEG.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.
• Aki, K., Richards, P. G. (1980). Quantitative Seismology: Theory and Methods. W. H. Freeman.
• Rutherford, S. R., Williams, R. H. (1989). Amplitude-versus-offset variations in gas sands. Geophysics, 54, 680-688.
• Ross, C. P., Kinman, D. L. (1995). Nonbright-spot AVO: Two examples. Geophysics, 60, 1398-1408.
• Castagna, J. P., Swan, H. W. (1997). Principles of AVO crossplotting. The Leading Edge, 16, 337-342.
• Castagna, J. P., Swan, H. W., Foster, D. J. (1998). Framework for AVO gradient and intercept interpretation. Geophysics, 63, 948-956.